# A Tandem Axial-Piston Unit Based Strategy for the Reduction of Noise Sources in Hydraulic Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Solution Analysis

_{1}for unit 1 and the sum of L

_{1}and L

_{2}for unit 2. This is also shown in Figure 1. The wavenumber k is dependent on the speed of sound c and angular frequency ω.

_{1}and B

_{2}are a characteristic of the flow sources, and they depend on the unit speed and a given wave frequency. Generally, these constants are empirically calculated using consolidated techniques, such as the “Secondary Source method” [37] and “Two pressure two system method” [38]. However, assuming the ideal case of two units being identical, the constants are strongly determined by the unit geometry. These constants could be assumed to be equal in both units. Thus, the pressure oscillation generated by each unit at the branch is represented by Equations (2) and (3):

_{2}, and the fixed phase shift ψ in between both units starting rotating point. Of these factors, the most critical is the speed of sound, which depends on several parameters, such as oil temperature, fluid pressure, the elastic properties of the duct material. This makes the phase too fickle to be controlled in different operating conditions unless there is no difference in length between the lines. In the specific case of perfectly symmetric lines, the phase difference between the waves is just the fixed phase shift, achieving the pattern of coherent interference. If the phase transformation is equivalent to 180°, the interference also becomes destructive, as shown by Equations (4) and (5). The elimination of the value L

_{2}and attribution of the value $\mathsf{\pi}$ to the phase shift $\mathsf{\psi}$ makes the pressure wave generated by unit 2 has the negative value of the pressure wave generated by unit 1 on the branch. This property holds to every position after the branch as well.

## 3. Simulation

_{b}is present.

#### 3.1. Axial Piston Unit Model

_{SBi}, Q

_{SKi}and Q

_{SGi}occur at their respective interfaces between the cylinder block and the valve-plate, between the piston and the cylinder block, and between the slipper and swash-plate.

_{ri}consists of the addition of flow between the given chamber and high-pressure port Q

_{rHPi}and low-pressure port Q

_{rLPi}(8). These flows are calculated by the orifice Equations (9) and (10), which are derived from Bernoulli’s equation by assuming an incompressible fluid. The respective orifices A

_{rHPi}, and A

_{rLPi}will be equivalent cross-sectional opening areas between the displacement chamber pressure ports and the unit valve-plate in a given moment of time. Since valve-plate design affects effective flow, cylinder pressure, and pressure pulsation, it had been studied extensively for noise reduction in axial-piston machines [20,21,22,23].

#### 3.2. Line Model

^{+}and C

^{-}are the pressure waves traveling at the speed of sound on positive and negative axial directions, respectively. From each curve, two ordinary differential Equations (17) and (18) can be deduced. Further details can found in References [44,45].

_{L}is called the isothermal wave speed.

## 4. Experimental Setup

_{1}, p

_{2}, and p

_{3}are respectively positioned 55, 335, and 408 cm away from the pump outlet.

## 5. Results

#### 5.1. Simulation Results

^{1/3}for measured geometric distances, 2

^{2/3}for geometric areas, and 2 for geometric volumes. The valve-plate area also was rescaled accordingly. To isolate the effects of the source generation, the larger unit simulation kept an extra branch with the close end in the simulation. The conduit diameter was kept constant as 25.6 mm through the whole line extension.

#### 5.2. Experimental Results

_{3}is installed. At 1200 rpm and 1660 rpm, respectively, the first harmonic was reduced by 15.32 and 17.96 dB at p

_{1}and by 16.87 and 15.87 dB at p

_{3}. Regarding the third harmonic, at 1200 rpm and 1660 rpm, respectively, the first harmonic was reduced by 10.31 and 9.55 dB at p

_{1}and by 11.63 and 9.3 dB at p

_{3}.

_{1}and pipe acceleration A

_{1}was calculated, just as the transfer function from pressure ripple p

_{3}to pipe wall acceleration A

_{2}. To represent the transfer function from input to output for one given unit speed, the value of the four operation condition power spectral density is averaged. This transfer function is also calculated by using the average of 20 measurements dividing the power spectral density of the input by the power spectral density of the output (25). This metric was called the transfer function ratio.

_{1}, the conduit is a tee, while in the position of p

_{3}, the conduit is a thin-walled pipe. Regardless of shape differences, both positions transfer functions show a critical frequency band from 2000–3000 Hz where amplifications up to 40 dB higher than the lower frequencies, the region where strategy proved to be effective. This frequency range is also the region where measured sound power displayed in Figure 12 had large peaks.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbols | Description | Unit |

A_{L} | Area of the duct | [m^{2}] |

A_{rHPi} | Valve plate area open to discharge port | [m^{2}] |

A_{rLPi} | Valve plate area open to suction port | [m^{2}] |

B | Constant | [-] |

B_{L} | Isothermal wave speed | [kg/m^{4}·s] |

C | Characteristic equation (used in MOC line model) | [Pa] |

c | Speed of sound | [m/s] |

D | Line diameter | [m] |

F | Force | [N][kg·m/s^{2}] |

F_{rki} | Displacement chamber to piston force | [N][kg·m/s^{2}] |

f_{DW} | Darcy–Weisbach friction coefficient | [m/s] |

f | Frequency | [Hz] |

I | Rotation kit index angle | [°] |

i | Displacement Chamber number | [-] |

j | Node | [-] |

K | Fluid bulk modulus | [Pa][kg/m·s^{2}] |

k | Spatial frequency (wave number) | [-] |

L | Line length | [m] |

M | Torque | [N·m] |

M_{x} | Swash plate moment about X axis | [Nm] |

M_{y} | Swash plate moment about Y axis | [Nm] |

M_{z} | Swash plate moment about Z axis | [Nm] |

m | Unit’s piston number | [-] |

N_{h} | Multiple of unit’s fundamental frequency | [-] |

n | Rotational speed | [rpm] |

O | Total number of branch segments | [-] |

o | Total number of branch segments | [-] |

p | Pressure | [Pa][kg/m·s^{2}] |

p_{i} | ith displacement chamber pressure | [Pa][kg/m·s^{2}] |

p_{j} | jth node pressure | [Pa][kg/m·s^{2}] |

p_{HP} | High pressure port pressure | [bar] [Pa][kg/m·s^{2}] |

p_{LP} | Low pressure port pressure | [bar] [Pa][kg/m·s^{2}] |

Q | Flow rate | [m^{3}/s] |

Q_{s} | Volumetric loss flow rate | [m^{3}/s] |

Q_{SBi} | Gap flow through VP and CB | [m^{3}/s] |

Q_{SGi} | Gap flow through slipper and swash plate | [m^{3}/s] |

Q_{SKBGi} | Total flow from the gaps | [m^{3}/s] |

Q_{SKi} | Gap flow through piston and cylinder block | [m^{3}/s] |

Q_{rHPi} | Flow from HP port to DC | [m^{3}/s] |

Q_{rLPi} | Flow from LP port to DC | [m^{3}/s] |

R_{b} | Cylinder block pitch radius | [m] |

S | Power Spectral Density | [bar/Hz] |

T | temperature | [C°] |

t | time | [s] |

V_{i} | Derived displacement chamber volume | [m^{3}] |

z | Position | [m] |

v | Fluid velocity | [m/s] |

α_{D} | Orifice coefficient of discharge | [-] |

β | Unit displacement | [°] |

λ | Wave length | [m] |

Δp | Pressure ripple | [bar] [Pa][kg/m·s^{2}] |

τ | Shear Stress | [Pa][kg/m·s^{2}] |

ρ | Density | [kg/m^{3}] |

ω | Angular velocity | [rad/s] |

ψ | fixed phase shift | [rad] |

φ_{i} | Piston angular position | [rad] |

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**Figure 12.**Time-domain simulation results at 100 bar 1200 rpm; at (

**a**) M

_{z}-Torque ripple at the shaft and (

**b**) pressure ripple at the branch, (

**c**) pressure ripple at the discharge.

**Figure 13.**Spectral power ratio frequency response function: 20° Index normalized over 0° index (

**a**) n = 1200 rpm; (

**b**) n = 1660 rpm.

**Figure 14.**Transfer Function Ratio of pressure ripple vs. pipe wall acceleration n = 1660 rpm at (

**a**) branch; (

**b**) pipe downstream.

Tag | Name | Sensor | Range |
---|---|---|---|

p_{s} | Pressure Source | Hydac | 60 bar |

p_{au} | Pressure A (HP) | Wika | 550 bar |

p_{d} | Pressure Drain | Keller | 30 bar |

p_{ad} | Pressure A (LP) | Wika | 25 bar |

p_{b} | Pressure B | Wika | 100 bar |

Q_{a} | Flow HP | VSE | 250 L/Min |

Q_{s} | Source Drain | VSE | 40 L/Min |

M | Torque | Staiger-Mohilo | 0–500 N m |

n | Speed | Staiger-Mohilo | 0–12,000 rpm |

ß_{1} | Displacement U1 | In-Built | 4–20 mA |

ß_{2} | Displacement U2 | In-Built | 4–20 mA |

T_{au} | Temp. A (HP) | Omega | K-type Thermocouple |

T_{s} | Temp. Source | Omega | K-type Thermocouple |

T_{d} | Temp. Drain | Omega | K-type Thermocouple |

T_{ad} | Temp. A (LP) | Omega | K-type Thermocouple |

T_{amb} | Temp. Ambient | Omega | K-type Thermocouple |

p_{1} | Pressure Ripple 1 | Kistler | −30–30 bar variation |

p_{2} | Pressure Ripple 2 | Kistler | −30–30 bar variation |

p_{3} | Pressure Ripple 3 | Kistler | −30–30 bar variation |

m_{1} | Microphone | G.R.A.S 40AO | 5 Hz to 12.5 kHz |

A_{1} | Accelerometer 1 | PCB 356A16 | ±50 g |

A_{2} | Accelerometer 2 | PCB 356A16 | ±50 g |

A_{3} | Accelerometer 3 | PCB 356A16 | ±50 g |

A_{4} | Accelerometer 4 | PCB 356A16 | ±50 g |

U1 Speed (rpm) | High Pressure (bar) | Displacement (%) | |
---|---|---|---|

1 | 1200 | 100 | 100 |

2 | 1200 | 175 | 100 |

3 | 1200 | 250 | 100 |

4 | 1660 | 100 | 100 |

5 | 1660 | 175 | 100 |

6 | 1660 | 250 | 100 |

7 | 1200 | 100 | 50 |

8 | 1660 | 100 | 50 |

Speed (rpm) | 1200 | 1660 | ||||||
---|---|---|---|---|---|---|---|---|

Pressure (bar) | 100 | 100 | 170 | 250 | 100 | 100 | 170 | 250 |

Disp. (%) | 50 | 100 | 100 | 100 | 50 | 100 | 100 | 100 |

0° Index | 83.9 dB | 86.7 dB | 92.0 dB | 94.0 dB | 85.4 dB | 84.8 dB | 89.1 dB | 91.8 dB |

20° Index | 82.9 dB | 84.3 dB | 89.4 dB | 92.7 dB | 85.1 dB | 85.9 dB | 90.3 dB | 93.6 dB |

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**MDPI and ACS Style**

Danes, L.; Vacca, A.
A Tandem Axial-Piston Unit Based Strategy for the Reduction of Noise Sources in Hydraulic Systems. *Energies* **2020**, *13*, 5377.
https://doi.org/10.3390/en13205377

**AMA Style**

Danes L, Vacca A.
A Tandem Axial-Piston Unit Based Strategy for the Reduction of Noise Sources in Hydraulic Systems. *Energies*. 2020; 13(20):5377.
https://doi.org/10.3390/en13205377

**Chicago/Turabian Style**

Danes, Leandro, and Andrea Vacca.
2020. "A Tandem Axial-Piston Unit Based Strategy for the Reduction of Noise Sources in Hydraulic Systems" *Energies* 13, no. 20: 5377.
https://doi.org/10.3390/en13205377